ISSN 0101-8205 www.scielo.br/cam
Study of exact solutions of nonlinear heat equations
A. EBADIAN and P. DARANIA
Department of Mathematics, Faculty of Science, Urmia University, P.O. Box 165, Urmia, Iran E-mails: a.ebadian@mail.urmia.ac.ir / p.darania@mail.urmia.ac.ir
Abstract. One of the ways that energy transports in fluid is electron thermal conduction. The aim of Inertial Confinement Fusion (ICF) is to show that the thermal conductivity is strongly dependent on temperature and the equation of heat condition is a nonlinear equation. In this article, we analyze the exact solutions of the nonlinear equation of heat conduction problem with variable transfer coefficients which is the problem of ICF.
Mathematical subject classification: 35K05, 35A20.
Key words:discrete group transformations, heat equation, Emden-Fowler equations.
1 Introduction
Inertial confinement fusion is a process where nuclear fusion reactions are ini-tiated by heating and compressing a target – a pellet that most often contains deuterium and tritium by the use of intense laser or ion beams. The beams ex-plosively detonate the outer layers of the target, accelerating the remaining target layers inward and sending a shock wave into the center. If the shock wave is powerful enough and if high enough density at the center is achieved some of the fuel will be heated enough to cause fusion reactions, releasing energy. In a target which has been heated and compressed to the point of thermonuclear ignition, energy can then heat surrounding fuel to cause it to fuse as well, creating a chain reaction that burns the fuel load, potentially releasing tremendous amounts of energy. Theoretically, if the reaction completes with perfect efficiency (though this is a practically impossible feat), a small amount of fuel about the size of a
pinhead, or around 10 milligrams, is capable of releasing the energy equivalent to burning a barrel of oil (that is, chemically combining its hydrocarbon molecules with oxygen).
Fusion reactions combine lightweight atoms, such as hydrogen, together to form larger ones. Generally the reactions take place at such high temperatures that the atoms have been ionized, their electrons stripped off by the heat; thus, fusion is typically described in terms of “nuclei” instead of “atoms”.
Fusion reactions on a scale useful for energy production require a very large amount of energy to initiate in order to overcome the so-called Coulomb barrier or fusion barrier energy. Since the positively-charged nuclei are naturally repelling each other, this repulsive force must be overcome by providing some form of external energy. When this occurs, however, the reaction is a rather energetic one. Generally less energy will be needed to cause lighter nuclei to fuse, and when they do, more energy will be released. As the mass of the nuclei increase, there is a point where the reaction no longer gives off net energy – the energy needed to overcome the energy barrier is greater than the energy released in the resulting fusion reaction. This point occurs when iron nuclei are formed and is the cause of death in some massive stars. This phenomenon plays no role in laboratory induced fusion however, since the energy and temperature required to form iron nuclei are very large. Fusion of heavy nuclei is possible using particle accelerators and is the method used to form very heavy transuranic elements such as Roentgenium for instance, though the method of achieving this fusion of heavy elements is far removed from the methods used in large scale fusion reactions which are desired in tokamak or ICF fusion reactors.
removing neutrons increases the energy barrier.
In order to create the required conditions, the fuel must be heated to tens of millions of degrees, and/or compressed to immense pressures. The temperature and pressure required for any particular fuel to fuse is known as the Lawson criterion. These conditions have been known since the 1950s when the first H-bombs were built.
ICF experiments started in earnest in the mid-1970s, when lasers of the required power were first designed. This was long after the successful design of magnetic confinement fusion systems, and even the particularly successful tokamak design that was introduced in the early 1970s. Nevertheless, high funding for fusion research stimulated by the multiple energy crises during the 1970’s produced rapid gains in performance, and inertial designs were soon reaching the same sort of “below breakeven” conditions of the best magnetic systems.
One of the earliest serious attempts at an ICF design was Shiva, a 20-armed neodymium laser system built at the Lawrence Livermore National Laboratory (LLNL) that started operation in 1978. Shiva was a “proof of concept” design, followed by the NOVA design with 10 times the power. Funding for fusion re-search was severely constrained in the 80’s, but NOVA nevertheless successfully gathered enough information for a next generation machine whose goal was ig-nition. Although net energy can be released even without ignition (the breakeven point), ignition is considered necessary for a practical power system.
This approach shows that, under the discrete group transformation, the solution of transformed equations can be converted into the solution of the reference equation [1,2,5,6].
2 Preliminaries
It is well known that the theory of differential equations takes a central place among possible instruments for the modeling of different processes and phe-nomena. Generalized Emden-Fowler equations arise in many different fields of applied sciences such as fluid mechanics [5], chemical physics [8] and physics of plasma [9].
In the present paper we consider the setDof ordinary differential equations,
which is referred to as a class of equations, whose members D(x,y,a) ∈ D
are uniquely defined by a vectoraof parameters. The Discrete Group Methods (DG M) based on the new fundamental mathematical theory is fully developed in [5,6,10].
All the existing methods of exact solution of ordinary differential equations can be conditionally divided into two groups:
A) a search for transformation of the original ordinary differential equations in classDto some other class of ordinary differential equationsD1, which
belongs to one of standard classes ofO D Ehaving known solutions.
B) a search for transformation leaving original ordinary differential equation inD invariant, i.e. transformation into “itself”, that gives independent
information about the solution.
The DG M does not operate with a single equation as in applications of Lie method [5], but operates with a class of equationsD, depending on a vectoraof
parameters, containing the investigated equation, but contrary to the approachA
one consider the transformations of the given classDwhich are closed in itself
on a chosen class ofO D E.
Definition 2.1. The class of generalized Emden-Fowler equations is written as:
where the three-dimensional parameter vector a = (n,m,l) ∈ R3, and the parameter A is insignificant. The parameter subspace(n,m,0)defines the set of classical Emden-Fowler equations
yx x′′ = Axnym.
LetDbe a class ofO D E and
D(x,y,a)=0, (2)
be an equation in this class, wherea is a vector parameter. We shall seek the transformations gi that are closed in the class (2), i.e., they change only the
vectora
gi :D(x,y,a)→ D(t,u,bi) (3)
if eachgi has an inverse, then the collection{gi}defines a Discrete
Transforma-tion Groups (DGT) on the class (2).
Definition 2.2. An R F -Pair is an operation of consecutive raising and lowering the order of equation.
Now, we define the following R-operations and F-operations:
i) Termwise m-fold differentiation of the original equation, type R Dm.
ii) Termwise one or two-fold differentiation of original equation with respect to the independent variable, type accordinglyR X orR X2.
iii) The equation is an exact derivative of themth order: termwise integration
mtimes, typeF Im.
iv) The equation is autonomous, i.e., it does not conclude an independent variable in an explicit form, typeF X
F X :y′x =u(y), yx x′′ =uu′y.
v) The equation is homogeneous in the extended sense, type FU: the trans-formationx =et, y =uekt, with an appropriate choice ofk, leads to an
autonomous form followed by a transformationF X.
Definition 2.3. Abel equation of the second kind, which is applied in many problems of mechanics, physics and other sciences is written as (see [5] for details):
yyx′ =F1(x)y+F0(y).
In literature, there are two methods for searching discrete group transforma-tions, namely point transformations and Bäcklund transformations:
2.1 Point transformations
In the class of point transformations:
y = f(t,u), x =g(t,u), J = fugt−guft 6=0, (4)
two methods are effectively applicable: the direct method and the method based on Lie algorithm [6]. The direct method is based on the substitution of the transformation (4) into (2). Imposing the condition J 6= 0 leads to a partial differential equation with unknown functions f andg, which can be split with respect to some independent variables into the lower order differential equations inu andt. The application of the direct method for equation (1) allows us to find two point transformations:
r:(y=t,x =u), r :(n,m,l)→(m,n,3−l), r2=E, and for l =0:
s:
y = u t,x =
1
t
, s :(n,m,0)→(−n−m−3,m,0), s2= E, whereEis the identity transformation.
2.2 Bäcklund transformations
(1), we obtain the following transformationg:
g:(n,m,l) −→
1 1−l,−
n n+1,
2m+1
m y′
x =t
1 1−l,
y =(u′t)−
1
m,
x =u 1
n+1,
−→ u′
t = y−m,
u=xn+1, t=(yx′)1−l,
g3= E =(gr)2.
Now by changing the indices of the above expressions assuming the trans-formed equation as the original and conversely, we obtain a transformationg−1:
g−1:(n,m,l) −→
− m
m+1, 1
l−2,
n−1
n
y′x =u 1 2−l,
y =t 1
m+1, x =(u′
t) 1 n, −→
u′t =xn,
u =(y′
x)2−l,
t =ym+1.
2.3 Solvable equations
The class of generalized Emden-Fowler equation (1) admits a general group
D3(g,r). (For feather details see [5,6])
The groupD3(g,r), is valid for all valuesn, mandl, except for the singular pointm = −1,0; n = −1,0; l =1,2. The group D3(g,r)may be extended smoothly on these points (except for the trivial solvable pointsm,n =0) while retaining the groupD3(g,r)structure. The technique for constructing the smooth extension is described in [2,5].
Now, by extending the classical Emden-Fowler equations associated to the vector(1−m,m,0)described in [2,5] and using the operatorsg,r ands with the consideration of equation (1), we obtain the groupD6(g,r,s). This group is valid for every value ofmexcept the singular points 0,±1,2.
Differential equation Analytical solution
y′′x x = Ax−4y5 y(x)=x 1+1 3x
2−12
y′′x x = Ax y−4(y′)3 x = ysin1
y
y′′x x = Ax−4y y(x)=xsin 1
x
y′′x x = Ax y−43(y′) 11
5 y(x)=
h
3x−23 −1
i−32
y′′x x = Ax−56y− 1 2(y′)
5
4 y(x)=
1−335x 1 3
3
y′′x x = Ax−12y− 5 6(y′)
7
4 y(x)=3− 9
5(1−x3)3
y′′x x = Ax−43y(y′)45 y(x)=h1 3
x−23 +3i− 3 2
y′′x x = Ax5y−4(y′)3 y(x)= x−2−13−
1 2
Table 1.
3 Analysis of the method
In this section presents analysis of the relation between the exact solutions of classical Emden-Fowler equation and nonlinear equations of heat conduction.
3.1 The formulation of the problem
According to [3], we consider the following problem of heat conduction in the spherical coordinatesr,θ,φ:
∇ ∙(K(T)∇T)= ∂T
∂t , (5)
with the following initial and boundary conditions:
T(r,0)=T0, T(1,t)=g(t).
whereT =T(r,t)is the temperature, K the heat transfer coefficient, andtthe time. Assume that
K =k0T52
wherek0is constant.
Equation (5) can be written as
∂K
∂r br ∙
∂T
∂rbr + K r2
∂ ∂r
r2∂T
∂r
= ∂T
By substituting:
∂K
∂r =
5k0
2 T 3 2∂T
∂r ,
into (6), we obtain the following equation:
5k0
2 T 3 2
∂T
∂r
2
+2k0 r T
5 2∂T
∂r +k0T
5 2∂
2T
∂r2 =
∂T
∂t. (7)
Application of the separation of variables:
T(r,t)=R(r)P(t), (8)
gives:
∂T
∂t =R P
′
t,
∂T
∂r = P R
′
r,
∂2T
∂r2 = P R
′′
rr,
and hence equation (7) reduces to:
5k0
2 R
1 2R′2
r +2k0r
−1R32R′
r +k0R
3 2R′′
rr =P
−72P′
t = −λ
2, (9)
whereλis the separation parameter. As a result of this decomposition, we obtain the following defining system:
P−72P′
t = −λ
2,
5k0
2 R
1 2R′2
r +2k0r−1R
3 2R′
r+ k0R
3 2R′′
rr = −λ2.
(10)
Note that the first equation of this system yields the function P(t), in the following form:
P(t)=
5 2λ
2t +c1
−25
, (11)
wherec1is a constant.
We shall now apply the following R F-pair operations, as a consequence of which the second equation of the system (10) is transformed to the classical Emden-Fowler equation. Having applied the operation:
where
Rr′ =es(k−1) u′s+ku,
Rrr′′ =es(k−2) u′′ss+(2k−1)u′s+k(k−1)u,
on the second equation of the system (10), we have the following equation:
es
5k 2−2
u′′ss+5
2u −1u′2
s +(7k+1)u
′
s+
7k2
2 +k
u+λ 2 k0u
−32
=0, (12)
and if we setk = 4
5, then equation (12) reduces to the autonomous equation
u′′ss+5
2u −1u′2
s + 33 5 u ′ s + 76 25u+
λ2 k0u
−32
=0. (13)
Now, by using the raising order operation F X : u′
s = w(u), u′′ss = wwu′,
equation (13) reduces to the general Abel equation to the following form:
wwu′ = −3
2u −1
w2−33
5 w− 76 25u−
λ2
k0u
−32
, (14)
where the general coefficients of this Abel equation according to the definition 3, is as follows:
φ0(u)=0, φ1(u)=1,
ψ0(u)= −
λ2
k0u
−32
−76
25u, ψ1(u)= − 33
5 , ψ2(u)= − 5 2u
−1
.
In this position, we consider the following transformations:
E =exp
−
Z ψ
2(u)
φ1(u) du
=u52, ν =
φ0(u)
φ1(u) +w
E =u52w, (15)
and define:
F0(u) =
ψ0
φ1
−φ0ψ1
φ2 1
+φ 2 0ψ2
φ13
E2− λ 2 k0u
7 2 − 76
25u
6,
F1(u) =
d du φ0 φ1
+ ψ1
φ1
−2φ0ψ2 φ12
E−33
By applying the transformation (15) on the equation (14), we obtain:
ννu′ = −33
5 u 5 2ν−λ
2 k0u
7 2 −76
25u
6
. (16)
The equation (16) by using the transformations:
τ =τ (u)=
Z
F1(u)du= −33
5
Z
u52du= −66 35u
7 2,
and
ϕ(u)= F0(u) F1(u) =
76 165u
7 2 + 5λ
2
33k0u,
reduce to
νντ′ −ν= 5λ 2
33k0τ+
76 165τ
7
2, (17)
where
τ = −66
35u 7
2. (18)
Now, by introducing the new constantsnandAin the forms:
5λ2
33k0 = −
(n+2) n+ 92
2(n+2)+ 522
, 76
165 = A
5
4(n+2)+5
2
, (19)
and using the transformation:
f = Aτ52, z = 5+4(n+2) 5
ν
τ −
2(n+2)
5 , (20)
equation (17) reduced to the equation:
(z−z2+ f)fz′=
5
2z+n+2
f. (21)
(For details see [8]).
Finally, through the substitution:
f = Axn+2y52, z = x
yy
′
x, (22)
the equation (21) is transformed to the classical Emden-Fowler equation:
where Ais an insignificant parameter, A andn depends onλ andλhas been defined in (9) as a separation parameter. Note thatnis also a constant depending on the initial and boundary conditions. For some values of n, this equation is integrable as a consequence of which, we have been able to construct some solution for the nonlinear heat conduction problem.
3.2 Exact solutions of the original equation
In this section, our aim is to obtain the exact solutions of the second equation of system (10), using the effect of inverse of theR F-pair operators on the exact solutions of the Emden-Fowler equation (23). For this purpose, in equation (23)
letn = −13
2 , then this equation is reduced to the following equation:
yx x′′ = Ax−132 y72,
with parametric solution as:
x =αC1∗lN(θ,C
∗
2)=αC
∗l
1 E
−1
7 2
, y=βC1∗hM(θ,C
∗
2)=βC
∗h
1 θE
−1
7 2
, (24)
where
A= ±9
4α 9 4 β−
5
2, h = 9
2, l =
5 2,
E7 2 =
Z
1±θ92− 1
2dθ +C∗
2, N = E
−1
7 2
, M =θE−71 2
,
andC∗
1,C
∗
2, αandβare constants andA= f(α, β)is an insignificant parameter
which is fully discussed in [6], pp. 254–255. By using the parametric solutions (20), we have
yx′ = d y d x =
β αC
∗(h−l)
1 θ −
E7 2
d dθE72
!
, (25)
and substituting (24) and (251) in the transformation (22), we obtain the exact solution of equation (21) in the following form:
f = Aα−92β 5 2C
∗
5h−9l 5
1 θ
5 2E2
7 2
, z =1−
E7 2
θddθE7 2
From equation (26) and (20), we obtainτ as:
τ =α−95βC ∗5h−59l
1 θE
4 5 7 2
, (27)
and also using the relation (19), we reformulate the expression forz in (20) as follows:
z = 662k0
5λ2
18 65
ν
τ +
9
5. (28)
Now, by substituting transformations (26) and (27) into (28), we obtain the parametric solution of the equation (17) in the following form:
ν= B0α−95βC ∗5h−59l
1 λ
2θ −4
5 −
E7 2 θddθE7
2 ! E 4 5 7 2 , (29) where
B0= 325
688k0.
On the other hand, by using (27), we have:
θE
4 5 7 2
=α95β−1C ∗5h−59l
1 τ, (30)
and substituting (30) in (29), the equation (29) is reduced to the following equation:
ν(τ )= −B0λ2τ
4 5+
D2(τ ) τD1(τ )
, (31)
where
D1(τ ) = d E7
2
dθ |θ=α95β−1C∗( 5h−9l
5 )
1 E
−45 7 2
τ
,
D2(τ ) = α−95βC ∗
5h−9l 5 1 E 9 5 7 2 |
θ=α95β−1C∗( 5h−9l
5 )
1 E
−45 7 2
τ
.
Now, the solution of (16) by using the relations (18) and (31) can be obtained as:
ν(u)= −B0λ2u
4 5+
D2(u)
u D1(u)
where
u=
− 7
14k+2
2
7 τ27.
Also, by replacing (15) into (32), we obtain the following solution for equa-tion (14):
w(u)= −B0λ2u−32
4 5 +
D2(u)
u D1(u)
. (33)
Applying the operator
F X−1:s =
Z du
w(u)
on the equation (33), yields the exact solution of the equation (13) in the follow-ing form:
s= − 1 B0λ2
Z u3
2
4 5+
D2(u)
u D1(u)
du.
Now, if we assume that this equation is integrable, then we set:
s = 1
B0λ2F(u), (34)
where
F(u)= −
Z u3
2
4 5+
D2(u)
u D1(u)
du.
Hence, by using the inverse operator FU:r = es, R(r) = urk in equation (34), this equation reduces to the exact solution of the second equation of system (10), which is as follows:
R(r)=uek
1
B0λ2F(u), (35)
whereu is a function ofr. Finally, by substituting (11) and (35) in (8), we get the exact solution of equation (7) in spherical coordinate as follows:
T(r,t)= R(r)P(t)=
uek
1 B0λ2F(u)
aλ2t+c1−
2
5. (36)
Due to the symmetry of the problem with respect toθandφ, we may setφ= π2, θ = 0 and obtain the same initial and boundary condition as T(r,0) = T0,
4 Conclusion
By introducing the someR F-pair operation and series of transformations beside
DGT, the nonlinear heat conduction problem with variable transfer coefficient in applied physics can be transformed into the classical Emden-Fowler equation which maybe integrated using classical methods. Then, by using the inverse of this operations and transformations, the exact solution of Emden-Fowler equa-tion reduced to the exact soluequa-tion of the nonlinear heat conducequa-tion problem. This approach shows that, under theR F-pair operations and transformations, the so-lution of transformed equation can be converted into the soso-lution of the reference equation.
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